Evidence for (Infinitely Diverse) Non-Convex Mirrors Tristan - - PowerPoint PPT Presentation
Evidence for (Infinitely Diverse) Non-Convex Mirrors Tristan Hbsch @ Southeastern Regional Mathematical String Theory Meeting V-Tech University, Blacksburg VA; 2017.10.07 Departments of Physics & Astronomy and Mathematics,
Tristan Hübsch
@ Southeastern Regional Mathematical String Theory Meeting V-Tech University, Blacksburg VA; 2017.10.07
Departments of Physics & Astronomy and Mathematics, Howard University, Washington DC Department of Physics, Faculty of Natural Sciences, Novi Sad University, Serbia Department of Physics, University of Central Florida, Orlando FL
gCI
Geometry: AAGGL 2015 BH 2016/06 GvG 2017
Toric
Geometry: Textbooks†…
(C)NLSM
Prehistory 1980s
— a mindmap
Diffeo-Data ☛ H*(X,ℤ) ☛ Chern classes ☛ Chern numbers ☛ Yukawa κ[ωA,ωB,ωC] ☛ p1[ωA] Holo-Data ☛ H*(X) ☛ H*(X,T) ☛ H*(X, EndT) ☛ Yukawa κ[ϕa,ϕb,ϕc] Quantum Data ☛ A-discriminants ☛ B-discriminants ☛ Yukawas ☛ Instantons, GW Semiclassical Data ☛ phases ☛ phase-boundaries
GLSM
Analysis ☛ W 1993 ☛ MP 1995 … BH 2016/11 BH 2017/10?
Avoid the poles of Laurent polynomials
Today!
P h y s i c s P h y s i c s
Prehistory The Big Picture Laurent GLSModels Phases & Discriminants …and in the Mirror
3
“It doesn’t matter what it’s called, …if… it has substance.” S.-T. Yau
Classical Constructions
Complete Intersections
Ex.: (x–x1)2+(y–y1)2+(z–z1)2 = R12
(x–x2)2+(y–y2)2+(z–z2)2 = R22
Algebraic (constraint) equations …in a well-understood “ambient” (A)
Work over complex numbers
…& incl. “infinity” (e.g., ℂℙn’s)
For hypersurfaces: X={p(x) = 0} ⊂ A
Functions: [f(x)]X = [f(x) ≃ f(x) + 휆·p(x)]A Differentials: [dx]X = [dx ≃ dx + 휆·dp(x)]A Homogeneity: ℂℙn = U(n+1)/[U(1)×U(n)]
Differential r-forms on ℂℙn are all U(n+1)-tensors
5
Just like gauge transformations …with tensors
Gauged Linear Sigma Model (GLSM) — on the world-sheet
Several “matter” fields + several “gauge” fields
Several coordinate functions – equivalence relations
“Kinetic” part (∥[∂ + qX A]X∥2): KE + gauge-matter coupling “Potential” part (W(X)): PE (gauge-invariant), “F-terms” “Gauge” part (∥∂∧A∥2 + τ·(∂∧A)): “D-terms” & “F.-I. terms”
World-sheet matter & gauge symmetries are both complex
E.g.: (x1, x2, x3) ≃ (λq1 x1, λq2 x2, λq3 x3), λ ∈ ℂ*: ℙ2
(q1:q2:q3)
…makes sense if the fixed-point set is excised (forbidden) from (x1, x2, x3) ∈ ℂ3 …or considered as an alternate (separate) location.
Gauge symmetry “stratifies” the X-field-space
& |vacuum⟩ determined by min[W(X)]: hypersurface
8
⇒ spacetime
Superstrings = Framework for Models
Aμ ≃ Aμ + (∇
μ λ)
Consider S2 ≃ ℙ1: Need at least two (complex) coordinates:
Match (the exponents) near the equator: (+1)N = (–1)S Symmetry: ξ→λ+1ξ and η→λ–1η, with λ ∈ ℂ* = (ℂ ∖ {0}) Explicitly: λ = ei(α+iβ) = e–β ·eiα = (real) rescaling · phase-change
Toric Geometry
9
ξ η
(–1) (+1)
ξ +1 = η−1
“thickened” S1
usual gauge transformation
More complicated examples: S2 ⨯ S2
An entire 2nd sphere at every point of 1st Orthogonal ↔ linearly independent Top-dim cones ↔ coord. patches 2-dim (enveloping) polytope ↔ (ℂ) 2-dim. geometry
More complicated yet: “twisted” product
T wisted torus S1 ⨳ S1 (S1 “twists” about S1) (≃ crystal w/oblique lattice $).
Now ⨯ℂ: Hirzebruch (ℂ) surface, F1.$
“Slanting” (0,–1) → (–m,–1) the bottom vertex (& two cones) encodes the “twist” … Fm = m-twisted ℙ1-bundle over ℙ1. …and so on: 4 textbooks worth!
10
Toric Geometry
s p a n n i n g p
y g
Polytope Encoding
The polytope encodes the space …but also its symmetries:
Assign each vertex a (Cox) coordinate Read off cancelling relations
Defines two independent (gauge) symmetries
a GLSM w/gauge-invariant Lagrangian and | ground state ⟩ where KE = 0 = PE & (quantum) Hilbert space on it
11
x1 x2 x3 x4
1~ vx1 + 1~ vx2 + 0~ vx3 + 0~ vx4 = 0
(x1, x2, x3, x4) ' (λ1 x1, λ1 x2, λ0 x3, λ0 x4)
0~ vx1 + m~ vx2 + 1~ vx3 + 1~ vx4 = 0
(x1, x2, x3, x4) ' (λ0 x1, λm x2, λ1 x3, λ1 x4)
m=1
A Generalized Construction of Calabi-Yau Models and Mirror Symmetry arXiv:1611.10300 BH
2-torus in the Hirzebruch surface Fm:
“ Anticanonical” (Calabi-Yau, Ricci-flat) hypersurface in Fm Toric description
13
spanning polytope
The star-triangulation of the spanning polytope defines the fan of the underlying toric variety
BH
N ?
F3 (1,0) (0,1) (1,0) (3,1) (m,1)
ˆ e1=(1,0) ˆ e2=(0,1) ˆ e1=(1,0) 3ˆ e1ˆ e2=(3,1) (m,1)
1 2 3 4 NR ΣF3
—Proof-of-Concept—
& Non-Convex Mirrors
arXiv:1611.10300
non-convex for m>2 (…also, non-Fano for m>2)
F3
3, 1
The “standard” polar polytope is non-integral The “standard” polar of the polar is not the spanning polytope that we started with Is no good for mirror symmetry
14
BH
The spanning polytope ∆?
F3 ⊂ NR
σ1 σ2 σ3 σ4 ν1 φ1 ν2 φ2 ν3 φ3 ν4 φ4 ((∆?
F3)) = Conv(∆? F3)
6= ∆?
F3
—Proof-of-Concept—
& Non-Convex Mirrors
arXiv:1611.10300
(Δ
★)°:={u: ⟨u,v⟩≥ –1, v ∈ Δ ★}
(ν1) (ν2) (ν3) (ν4) (φ1) (φ2) (φ3) (φ4) The oriented Newton polytope (trans-polar of spanning polytope): Construction (trans-polar)
Decompose Δ
⭑ into
convex faces θi; Find the (standard) polar (θi)° for each (convex) face (Re)assemble parts dually to (θi ∩ θj)° = [(θi)°, (θj)°] with “neighbors” Agrees with standard (if obscure?) constructions…
15
BH —Proof-of-Concept—
& Non-Convex Mirrors
ν1 ν2 ν3 ν4 φ1 φ2 φ3 φ4
arXiv:1611.10300
“Normal fan” Dual cones ↦ inside opening vertex-cones trans-polar
The oriented Newton polytope:
specifies allowed monomials The so-defined 2-tori are all singular @(0,0,1) …as each monomial has at least an x1 factor, so f(x) = x1·g(x) The extension corresponds to Laurent monomials:
16
BH
(1,4) (1,1+m) (1,3) (1,2) (1,1) (1,0) (1,1) (0,1) (0,0) (0,1) ( 2
m ,1)! ( 2 3 ,1)
%
(1,1) (1,1m)! (1,2)
M F3 (?
F3)
F3
& Non-Convex Mirrors
arXiv:1611.10300
x 2
1 x 5 3
x 2
1 x 4 3 x4
x 2
1 x 3 3 x 2 4
x1 x2 x 2
3
x 2
1 x 2 3 x 3 4
x1 x2 x3 x4 x 2
1 x3 x 4 4
x1 x2 x 2
4
x 2
1 x 5 4
(1, 1) 7!
x 2
2
x4
(1, 2) 7!
x 2
2
x3
make the 2-tori Δ-regular.
9 = ;
2 6 6 6 4 2 5 2 5 2
1
2
1
3 7 7 7 5 2 6 6 6 4 2 2 2 2 5
1
5
1
3 7 7 7 5
—Proof-of-Concept— The oriented Newton polytope:
is star-triangulable → a toric variety differs from its convex hull by “flip-folded” simplices
Associating coordinates to corners:
SP: x1=(–1,0), x2=(1,0), x3=(0,1), x4=(–3,–1) NP: y1=(–1,4), y2=(–1,–1), y3=(1,–1), y4=(1,–2) Expressing each as a monomial in the others:
17
BHK
Mirror Construction arXiv:hep-th/9201014
BH
P2
(1:1:3)[5]
P2
(3:2:5)[10]
& Non-Convex Mirrors
arXiv:1611.10300
; NP: x 2
1 x 5 3 x 2 1 x 5 4 x 2 2
x4
x 2
2
x3 vs. SP: y 2
1 y 2 2 y 2 3 y 2 4 y 5 1
y4
y 5
2
y3
☛“multi-fans”
M a s u d a , + H a t t
i ’ 9 9 K a r s h
+ T
m a n ’ 9 3 K h
a n s k i i + P u k h l i k
’ 9 2
K3 in Hirzebruch 3-folds, “cornerstone” mirrors: The Hilbert space & interactions restricted by the symmetries
Analysis: classical, semi-classical, quantum corrections…
…in spite of the manifest singularity in the (super)potential
18
BH
a1 x8
4 + a2 x8 3 + a3
x3
1
x3 + a5 x3
2
x3 : exp 8 > > < > > : 2iπ 2 6 6 4
1 3 2 3 0 0 1 24 1 24 1 8 0 3 8 3 8 1 8 1 8
3 7 7 5 9 > > = > > ; " x1 x2 x3 x4 # : ⇢ G = Z3 × Z24, Q = Z8.
2 6 6 6 4 0 0 0 8 0 0 8 0 3 0 −1 0 0 3 −1 0 3 7 7 7 5
|G| |Q| = 3·24 8 = 9 = d(∆F3) d(∆?
F3) = 54
6 .
(3:3:1:1)[8]
P3
(3:5:8:8)[24]/Z3
—Proof-of-Concept—
& Non-Convex Mirrors
arXiv:1611.10300
× b1 y3
3 + b2 y3 5 + b3
y8
2
y3 y5 + b4 y8
1 :
exp 8 > > < > > : 2iπ 2 6 6 4
1 8
0 0 0
1 3 2 3 3 24 5 24 1 3 1 3
3 7 7 5 9 > > = > > ; 2 4 y1 y2 y3 y5 3 5 : ⇢ GO = Z8 × Z3, QO = Z24.
Z8 × Z3,
BHK
BH
The Phase-Space
20
The (super)potential: The possible vevs
W(X) := X0 · f(X), ✓ · f(X) :=
2
X
j=1
✓
n
X
i=2
i
n+j + aj Xn 1 X (n−1)m+2 n+j
◆ ,
X0 X1 X2 · · · Xn Xn+1 Xn+2 Q1 n 1 1 · · · 1 Q2 m2 m 0 · · · 0 1 1
|x0| |x1| |x2| · · · |xn| |xn+1| |xn+2| i · · · ⇤ ⇤ I ⇤ ⇤ · · · ⇤ ⇤ ⇤ ii ⇤ · · · ⇤ II see (2.9) ⇤ · · · ⇤ ⇤ ⇤ iii pr1 · · · III q
mr1+r2 (n−1)m+2
q
(m−2)r1+nr2 (n−1)m+2
· · · iv p r1/n · · · IV p r1/n · · · ⇤ ⇤
r1 r2 I II III IV
(1, 0) (0, 1) (−n, m−2) (1, −m)
(i ) (ii ) (iii )
⇣ ⇣ ⇣ )
(iv )
|x1| = sP j |xn+j|2 r2 m = v u u tr1 n X i=2 |xi|2 > 0arXiv:1611.10300
—Proof-of-Concept—
r1 r2 I II III IV
(1, 0) (0, 1) (−n, m−2) (1, −m)
(i ) (ii ) (iii )
⇣ ⇣ ⇣ )
(iv )
BH
21
Phase-space:
I: Calabi-Yau (n–1)-fold hypersurface in Fm II: Calabi-Yau (n–1)-fold hypersurface in Fm (flopped) III: Calabi-Yau ℤ(n–1)m+2 Landau
IV: Calabi-Yau hybrid
The ⟨xi(r1,r2)⟩ change continuously ’round (0,0); boundaries are singular only for special values of θ.
The Phase-Space
arXiv:1611.10300
—Proof-of-Concept—
BH Varying m in Fm:
The Phase-Space
arXiv:1611.10300
—Proof-of-Concept— F0 = ℙ1⨯ℙ1 F1 = B[ℙ1]pt. F2 F3 F4
W 1993 M&P 1995
22 (1,0) (0,1) (1,–m) (–n,2–m)
(2)
3 )
I II III IV
L1
L1
L1 ⇥ W (F
(2)
3 )
⇤ L1 L1
6= 6=
W (F
(2)
1 )
I II III IV
BH Infinite diversity in the Fm: The [m (mod n)] diffeomorphism
The Phase-Space
arXiv:1611.10300
—Proof-of-Concept—
23
| | e Lk : F
(n)
m [c1] ! F
(n)
m+nk[c1] d L1 :
W (F (n)
m [c1])
z }| { (1, 0), (n, m2) ·
h 1 n 0 1 i
6=
m−n[c1])
z }| { (1, 0),
L1 :
W (F (n)
m [c1])
z }| { (0, 1), (1, m) ·
h 1 n 0 1 i
m−n[c1])
z }| { (0, 1),
BH
24
arXiv:RealSoon
The Discriminant
Now add “instantons”: 0-energy string configurations wrapped around “tunnels” & “holes” in the CY spacetime
Near (r1,r2) ~ (0,0), classical analysis
fails [M&P: arXiv:hep-th/9412236]
With the instanton resummation gives: —Proof-of-Concept—
r1 + ˆ ✓1 2⇡i = 1 2⇡ log ✓ n−1
1
(1m 2) [(m2)2n1]n ◆ , r2 + ˆ ✓2 2⇡i = 1 2⇡ log ✓ 2
2 [(m2)2n1]m−2
(1m 2)m ◆ .
X0 X1 X2 · · · Xn Xn+1 Xn+2 Q1 n 1 1 · · · 1 Q2 m2 m 0 · · · 0 1 1
BH
26
arXiv:RealSoon
The Discriminant
—Proof-of-Concept—
Now compare with the complex structure of the BHK-mirror
Restricted to the “cornerstone” def. poly
In particular,
g(y) =
n+2
X
i=0
bi φi(y) = b0 φ0 + b1 φ1 + b2 φ2 + b3 φ3 + b4 φ4, φ0 := y1 · · · y4, φ1 := y 2
1 y 2 2 ,
φ2 := y 2
3 y 2 4 ,
φ3 := y m+2
1
y m−2
3
, φ4 := y m+2
2
y m−2
4
, z1 = −β [(m−2)β + m] m+2 , z2 = (2β+1)2 (m + 2)2 βm , β := b1 φ1 b0 φ0 .
A
J(g)
Identical with Kähler mirror
Batyrev
f(x) = a0 Y
νi2∆?
(xνi)hνi,µ0i+1 + X
µI2∆
aµI Y
νi2∆?
(xνi)hνi,µIi+1 Y
2
X
2
Y
2
g(y) = b0 Y
µI2∆
(yµI)hµI,ν0i+1 + X
νi2∆?
bνi Y
µI2∆
(yµI)hµI,νii+1
BH
27
arXiv:RealSoon
The Discriminant
—Proof-of-Concept—
So, and are identical?! Better yet:
M (O F (n)
m [c1]) :
8 > > > < > > > : z1 = (1)n1 β [(m2) β + m]n1 [(n1)m+2]n1 , z2 = (1 + n β)2 [(n1)m+2]2 βm , β := b1 φ1 b0 φ0 .
A
J(g)
∈ ∈ M (OF
(n)
m ) ⊃ P1 ≈
− − − − − − − − − → P1 ⊂ W (F
(n)
m )
γ := b3 φ3 b2 φ2 .
A
J(g)
= 1 m γ [(m2)γ 2]2 , z2 = γ2 [(m2)γ 2]m−2 (1 m γ)m , W (F (n)
m ) :
e−2πr1+iˆ
θ1 =
1 − m ρ [(m−2)ρ − n]n , e−2πr2+iˆ
θ2 = ρ2 [(m−2)ρ − n]m−2
(1 − m ρ)m ; ρ := σ2 σ1 . (σ1, σ2)
MM
≈ (b2φ2, b3φ3)
J
Horn uniformization Morrison+Plesser, ’93
" b0 b1 # = " −n m−2 1 −m # " σ1 σ2 #
You bet:
BH
28
arXiv:RealSoon
The Discriminant
—Proof-of-Concept—
So: In fact, also:
…when restricted to no (MPCP) blow-ups & “cornerstone” polynomial
Then, Same method:
), W (F
(n)
m [c1])
mm
≈ M (O F
(n)
m [c1]). O
mm
≈ at W (O F
(n)
m [c1])
mm
≈ M (F
(n)
m [c1]).
dim W (OF (n)
m [c1]) = n = dim M (F (n) m [c1])
e2πi e
τα = 2n
Y
I=0
✓
2
X
β=1
e Qβ
I e
σβ ◆ e
Qα
I
I
P
β e
Qβ
I e
σβ
J
(210)(f)
2(m+2)(e σ1 + e σ2) 2
m e σ1 + 2 e σ2
m (a3 ϕ3)+2 (a4 ϕ4) m+2
2 2 e σ1 + m e σ2
2 (a3 ϕ3)+m (a4 ϕ4) m+2
3 (m+2) e σ1 (a3 ϕ3) 4 (m+2) e σ2 (a4 ϕ4)
˜ za =
2n
Y
I=0
e
Qα
I . A
J
✅
BH —Proof-of-Concept—
Summary
arXiv:1611.10300 + more
CY(n–1)-folds in Hirzebruch 4-folds
Euler characteristic Chern class, term-by-term Hodge numbers Cornerstone polynomials & mirror Phase-space regions & mirror Phase-space discriminant & mirror The “other way around” Yukawa couplings World-sheet instantons Gromov-Witten invariants
Will there be anything else?
30
BH —Proof-of-Concept— ✅ ✅ ✅ ✅
d(θ (k)) := k! Vol(θ (k)) [BH: signed by orientation!]
Oriented polytopes Trans-polar▿ constr. Newton ΔX := (Δ
⭑ X)▿
VEX polytopes s.t.: ((Δ)▿)▿ = Δ Star-triangulable w/flip-folded faces Polytope extension ⇔ Laurent monomials (B)BHK mirrors
Summary
arXiv:1611.10300 + more
✅ ✅ Textbooks to be (re)written, amended
✅ ✅ *✅ ✅ (limited)
http://physics1.howard.edu/~thubsch/
Departments of Physics & Astronomy and Mathematics, Howard University, Washington DC Department of Physics, Faculty of Natural Sciences, Novi Sad University, Serbia Department of Physics, University of Central Florida, Orlando FL